Parabola practice · Easy

Option B (${y_1},{y_3},{y_2}$ are in AP) is correct because the ordinate of the intersection point of tangents to a parabola at two points is the arithmetic mean of the ordinates of those points.

Option A (5) is correct because the locus of the midpoints of parallel chords of a parabola is a line parallel to the axis of the parabola, and comparing with the given equation yields k=5.

Option B (The line through Q parallel to y-axis) is correct because the tangents at P and R intersect on a vertical line passing through Q, given that the ordinates are in geometric progression.

Option C ($\frac{3}{2} \sqrt{41}$) is correct because it is the calculated length of the chord of contact using the given point and parabola equation.

Option B ((1,2) and (4,4)) is correct because these are the points of contact of the tangents from P(2,3) to the parabola y²=4x.

Option B (9) is correct because it is the value of 'k' when the given line is normal to the parabola.

Option A. $C_1$ and $C_2$ touch each other at one point is incorrect. The correct answer is B ($C_1$ and $C_2$ touch each other exactly at two points) because the curves intersect at two points and have the same tangent at each point.

Option A ($h > 2$) is correct because for three distinct normals to be drawn from a point on the axis, the x-coordinate of the point must be greater than 2.

Option B ($x=2 t^2, y=4 t$) is correct because substituting these parametric equations into $y^2=8x$ yields a true statement.

Option A ($x+2y=4$) is correct because it represents the directrix, which is perpendicular to the axis and equidistant from the vertex as the focus.

Option A ($x = \pm \frac{{{a^2}}}{{\sqrt {{a^2} - {b^2}} }}$) is correct because the locus of the intersection of tangents at the endpoints of a focal chord is the directrix of the ellipse.

Option D ($y=x+2$) is correct because by finding the tangent to the parabola and then applying the condition for tangency to the hyperbola, we arrive at this equation.

Option B (0) is correct because the sum of the ordinates of the feet of the normals from any point to the parabola ${y^2} = 8x$ is zero.

Option B ((3,0)) is correct because the normals at the ends of the latus rectum of the parabola $y^2=4x$ intersect at the point (3,0).

Option C (-128) is correct because it is the value of b that satisfies the condition that the line y=mx+4 is a common tangent to both parabolas.

Option C (${\tan ^{ - 1}}\frac{1}{3}$) is correct because it is the angle between the tangents drawn from the given point to the parabola.

Option C (R) is correct because the condition for the tangent to be normal to the circle is satisfied for all real values of 'a'.

Option C (x-y-1=0) is correct because it represents the locus of point P, derived from the given condition on the inclinations of the tangents.

Option D (($\frac{{-1}}{{2}}, 0$) $\cup$ ($0, \frac{{1}}{{2}}$)) is correct because it defines the range of 'a' for which three normals can be drawn from the given point to the parabola.

Option A (x + y = 3) is correct because it is the equation of the normal to the parabola that passes through the given point (3,0).

Option B ($3 \sqrt{2} a$) is correct because the length of the common tangent is calculated by finding the points of tangency and then using the distance formula.

Option B A parabola whose vertex is $\left( {3,4} \right)$ and directrix is $x = \frac{{11}}{4}$ is correct because the given equation can be rewritten in the standard form of a parabola, and its vertex and directrix can be determined.

Option C ($( \pm 2,2)$) is correct because it minimizes the distance between the point and the parabola.

Option B ($a \in (0,4)$) is correct because the point must lie inside both the parabola and the region bounded by the double ordinate.

Option D ($2y - 3 = 0$) is correct because it represents the equation of the directrix of the given parabola, calculated using the standard form of a parabola.

Option B (16) is correct because for a focal chord of a parabola, the product of the x-coordinates of the endpoints is equal to $a^2$, where $4a$ is the coefficient of $x$ in the parabola's equation.

Option C ($a>1 / 2$) is correct because for three normals to be drawn to the parabola, the condition $a>1/2$ must be satisfied.

Option A ($x^2+4 y^2+4 x y+116 x+2 y+259=0$) is correct because it satisfies the definition of a parabola: the distance to the focus equals the distance to the directrix.

Option B ($y = - a$) is correct because the locus of the point of intersection of perpendicular tangents to a parabola is its directrix.

Option B ($(1,2 \sqrt{3}),(1,-2 \sqrt{3})$) is correct because substituting the focal distance into the parabola equation yields these points.